Set-Consensus using Set-Valued Observers

Set-Consensus using Set-Valued Observers

2015 American Control Conference Chicago, Illinois, USA.

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Outline

Motivation

Distributed Sensing - Each node computes estimates and need to synchronize them before

aggregating them.

Robot Coordination - Fleet of robots wishes to agree on direction/speed or rendezvous point.

Asynchronous Algorithms - Nodes acting independently changes the number of considered time steps as

1 2 5 4 3 Laser, RF, etc. communications tower Camera, ultra-sound transducer, IR sensor, or other

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Set Consensus

A group of n nodes is trying to achieve consensus.

Nodes have neither sensing nor self-localization capabilities. A tower uses a directional antenna to transmit to the nodes their position and velocity.

Two main issues: measurements are corrupted by noise and taken at different time instants.

Motivating Example

Consider the case of two vehicles given in the figure.

Node 1 receives the last measurement at

time instant k_{− 4.}

Due to sensor noise or disturbances, node 1 has only access to estimates.

Then, the decision might result in a collision!

v

(k)

p

(k)

p

(k − 4)

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Motivating Example

Consider the case of two vehicles given in the figure.

Node 1 receives the last measurement at

time instant k_{− 4.}

Due to sensor noise or disturbances, node 1 has only access to estimates.

Then, the decision might result in a collision!

v

(k)

p

(k)

ˆv

(k − 4)

Motivating Example

Consider the case of two vehicles given in the figure.

Node 1 receives the last measurement at

time instant k_{− 4.}

Due to sensor noise or disturbances, node 1 has only access to estimates.

Then, the decision might result in a collision!

v

(k)

p3(k)

ˆv3(k − 4)

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Problem Outline

Take n nodes, where each node i has dynamics of the form

xi(k + 1) = Ai(k)xi(k) + Bi(k)ui(k) + Ei(k)di(k)

ui(k) is the actuation signal and di(k) possible disturbances.

Set-Consensus Problem

How to achieve position or velocity consensus when instead of knowingxi(k) only a set Xi(k) is known such that xi(k) ∈ Xi(k).

Problem Model

Each agent i has a system of the form

xi(k + 1) =  A0+ n∆ X `=1 ∆`(k)A`  xi(k) + Bi(k)ui(k) + Eidi(k)

Each Si is a Linear Parameter-Varying (LPV) system

n∆ number of uncertainties

∆`(k) are scalar uncertainties with |∆`(k)| ≤ 1

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks SVO

Proposed Solution

Broadcast Solution using Position

Use Set-Valued Observers (SVOs) to update the received Xj(k − kj) for each of the neighbors j;

Compute the weighted average of the updated Xj(k);

Compute the velocity vector to drive Xi(k) to Xavg(k).

Unicast Solution using Estimation

Nodei receives sets Xj(k − kj) from a subset of its neighbors;

SetXi(k) will include the concatenation of the updated Xj(k)

and disturbance terms to account for each node j actuation;

The velocity vector will take into account the estimated position and velocity of the neighbors.

SVOs

Given the previous setX(k):

Using SVOs, the algorithm predicts ˜X(k + 1) using the

dynamics;

Then, the set is intersected with the measurement set Y (k + 1).

X(k)

X(k + 1)

˜

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks SVO

Algorithm

Velocity vector v can be found through:

v = arg min max

x,y (||(v + x) − y||)

subject to x_{∈ X}i(k)

Properties

Nodes position converge to a ball of radius equal to the maximum uncertainty in the measurement sets;

For the case of unicast communication and using estimates, uncertainty is higher as there are added disturbances and dynamics uncertainties in the update of the estimates; Convergence for a single cluster depends on the allocation of transmissions by the various directions.

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Simulation Results (1/2)

Setup: 200-node network randomly distributed over a 50m×50m

square and round-robin service using an offset to cover 10 partitions of the terrain.

In a typical run nodes converge to a smaller number of clusters (5 in the example).

Nodes aligned themselves along the partitions.

Figure depicts the evolution of the maximum distance between any two nodes.

Convergence to a cluster can be identified when there is little oscillation in this metric.

0 5 10 15 20 25 30 35 40 45 50 21 22 23 24 25 26 27 meters(m) meters(m)

Simulation Results (1/2)

Setup: 200-node network randomly distributed over a 50m×50m

square and round-robin service using an offset to cover 10 partitions of the terrain.

In a typical run nodes converge to a smaller number of clusters (5 in the example).

Nodes aligned themselves along the partitions.

Figure depicts the evolution of the maximum distance between any two nodes.

Convergence to a cluster can be

40 45 50 55 60 65 70 distance(m)

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Simulation Results (2/2)

Setup: 200-node network randomly distributed over a 50m_×50m

square with two antennae (length and width) used in a periodic scheduling. Amround-robin service is used for each antenna using an offset to cover 10 partitions of the terrain.

A typical run achieves consensus for a single cluster.

The maximum ball around the nodes has radius equal to the

maximum uncertainty max.

The maximum difference between two nodes converges to a value

smaller thanmax. 24.525.5 26 26.5 27 27.5 28 25 25.5 26 26.5 27 meters(m) meters(m)

Simulation Results (2/2)

Setup: 200-node network randomly distributed over a 50m_×50m

square with two antennae (length and width) used in a periodic scheduling. Amround-robin service is used for each antenna using an offset to cover 10 partitions of the terrain.

A typical run achieves consensus for a single cluster.

The maximum ball around the nodes has radius equal to the maximum uncertainty max. The maximum difference between

two nodes converges to a value 10

20 30 40 50 60 70 distance(m)

Introduction Problem Statement Proposed Solution Main Properties Simulation Results Concluding Remarks

Concluding Remarks

the use of SVOs to update the set representing the uncertainty about the position of the nodes; Two scenarios are addressed:

Broadcast - nodes use the positions for the other nodes; Unicast - nodes obtain information in the shared medium and estimate the position for the other nodes.

the positions of the nodes are shown to converge to the vicinity of the remaining nodes dependent on a measure of the uncertainty.

In Simulation, it is observed that the policy for the communication influences the number of clusters.

Set-Consensus using Set-Valued Observers

2015 American Control Conference Chicago, Illinois, USA.